Real-time forecasting US GDP from small-scale factor models

Abstract

We show that the single-index dynamic factor model developed by Aruoba and Diebold (Am Econ Rev, 100:20–24, 2010) to construct an index of the US business cycle conditions is also very useful to forecast US GDP growth in real time. In addition, we adapt the model to include survey data and financial indicators. We find that our extension is unequivocally the preferred alternative to compute backcasts. In nowcasting and forecasting, our model is able to forecast growth as well as AD and better than several baseline alternatives. Finally, we show that our extension could also be used to infer the US business cycles very precisely.

Appendix

Without loss of generalization, we assume that our model contains only GDP, one non-financial monthly indicator, and one financial monthly indicator, which are collected in the vector $Y_{t} =\left( {y_{t}^{*} ,Z_{it}^{*},Z_{ft}^{*}} \right) ^{{\prime }}$. For simplicity’s sake, we also assume that $p1 = p2 = p3 = 1$, and that the lead for the financial indicator is $h$ = 1. In this case, the observation equation, $Y_{t}=Z\alpha _{t}$, is

$$\begin{aligned} \left( {{\begin{array}{l} {y_{t}^{*}} \\ {Z_{it}^{*}} \\ {Z_{ft}^{*}} \\ \end{array}}} \right) =\left( {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} 0&{} {\frac{\beta _{y}}{3}}&{} {\frac{2\beta _{y}}{3}}&{} {\beta _{y}}&{} {\frac{2\beta _{y}}{3}}&{} {\frac{\beta _y }{3}}&{} {\frac{1}{3}}&{} {\frac{2}{3}}&{} 1&{} {\frac{2}{3}}&{} {\frac{1}{3}}&{} 0&{} 0 \\ 0&{} {\beta _{i}}&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 1&{} 0 \\ {\beta _{f}}&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 1 \\ \end{array}}} \right) \left( {{\begin{array}{c} {x_{t+1} } \\ {x_t } \\ \vdots \\ {x_{t-4} } \\ {u_t^y } \\ \vdots \\ {u_{t-4}^{y}} \\ {u_t^i } \\ {u_t^f } \\ \end{array}}} \right) .\nonumber \\ \end{aligned}$$

(20)

It is noteworthy that the model assumes contemporaneous correlation between non-financial indicators and the state of the economy, whereas for financial variables, the correlation is imposed between current values of the indicators and future values of the common factor.

The transition equation, $\alpha _{t} =T\alpha _{t-1} +\eta _{t}$, is

$$\begin{aligned} \left( {{\begin{array}{c} {x_{t+1} } \\ {x_{t}} \\ \vdots \\ {x_{t-4} } \\ {u_{t}^{y}} \\ \vdots \\ {u_{t-4}^{y}} \\ {u_{t}^{i}} \\ {u_{t}^{f}} \\ \end{array} }} \right) =\left( {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} {\uprho _{1}}&{} \cdots &{} 0&{} 0&{} 0&{} &{} \cdots &{} &{} &{} 0 \\ 1&{} &{} &{} 0&{} &{} &{} \cdots &{} &{} &{} 0 \\ \vdots &{} \ddots &{} &{} \vdots &{} &{} &{} \cdots &{} &{} &{} \vdots \\ 0&{} \cdots &{} 1&{} 0&{} &{} &{} \cdots &{} &{} &{} 0 \\ 0&{} \cdots &{} &{} 0&{} {d_{1}^{y}}&{} &{} 0&{} 0&{} &{} 0 \\ \vdots &{} \cdots &{} &{} \cdots &{} &{} \ddots &{} &{} &{} &{} \vdots \\ 0&{} \cdots &{} &{} &{} &{} &{} 1&{} 0&{} &{} 0 \\ 0&{} \cdots &{} &{} &{} &{} &{} 0&{} 0&{} {d_{1}^{i}}&{} 0 \\ 0&{} \cdots &{} &{} &{} &{} &{} 0&{} 0&{} 0&{} {d_{1}^{f}} \\ \end{array} }} \right) \left( {{\begin{array}{c} {x_t } \\ {x_{t-1} } \\ \vdots \\ {x_{t-5} } \\ {u_{t-1}^{y}} \\ \vdots \\ {u_{t-5}^{y}} \\ {u_{t-1}^{i}} \\ {u_{t-1}^{f}} \\ \end{array} }} \right) +\left( {{\begin{array}{c} {e_{t+1} } \\ 0 \\ \vdots \\ 0 \\ {\varepsilon _{t}^{y}} \\ \vdots \\ {\varepsilon _{t-4}^{y}} \\ {\varepsilon _{t}^{i}} \\ {\varepsilon _{t}^{f}} \\ \end{array} }} \right) ,\nonumber \\ \end{aligned}$$

(21)

where $\eta _t \,{\sim }\, iN \left( {0,Q} \right)$ and $Q={\textit{diag}}\left( {\sigma _{e}^{2} ,0, \ldots ,0,\sigma _{y}^{2} ,0 , \ldots , 0,\sigma _{i}^{2} ,\sigma _{f}^{2}} \right)$.